Dinamička modalna dekompozicija u empirijskoj identifikaciji nelinearnih dinamičkih sustava

master's thesis

• Author: Ivana Boras
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisujemo DMD metodu kao alat za numeričku analizu nelinearnih dinamičkih sustava. DMD metoda je metoda trenirana podacima i nije potrebno poznavati jednadžbu gibanja promatranog sustava, zbog tih poželjnih svojstava danas pronalazi primjenu u raznim područjima. Analiziramo matematičku pozadinu i vidimo da DMD metoda izdvaja važne strukture iz visoko dimenzionalnog dinamičkog sustava na prostorno i vremensko ovisne podatake. Te strukture su dane u svojstvenim vektorima operatora A koji opisuje promatranu dinamiku sustava. Operator A je općenito nepoznat, no pomoću prikupljenih uzoraka sustava, DMD odreduje linearnu aproksimaciju operatora A, matricu S r i traži svojstvenu dekompoziciju te matrice. To postiže koristeći SVD dekompoziciju matrice i Rayleigh-Ritzovu metodu za računanje svojstvene dekompozicije. Pokazujemo kako pomoću DMD algoritma rekonstruirati dinamički sustav. Zatim definiramo Koopmanove operatore i opisujemo djelovanje Koopmanovog operatora na funkcije stanja sustava. Dajemo kratak opis svojstvene dekompozicije Koopmanovih operatora te povezanost s DMD metodom. Pokazujemo primjenu DMD metode u procesiranju videa, tj. u odvajanju pozadine videa od prvog plana videa. Metodu testiramo na dvije baze, te dobivene rezultate usporedujemo s RPCA metodom koja je trenutno najrasprostranjenija u tom području. U zadnjem poglavlju dajemo varijacije DMD algoritma. Pokazujemo kako odabrati parametar redukcije dimenzije, kako poboljšati Rayleigh-Ritzove parove dobivene DMD algoritmom i tako opisujemo još dva nova algoritma: DDMD_R4 i DDMD_R4_C. Zatim, pokazujemo kako aproksimirati operator A ako želimo da nam prostor u kojem tražimo aproksimaciju razapinju odabrani uzorci stanja. Vidimo da tražene rezultate možemo postići ako umjesto SVD dekompozicije matrice koristimo QR dekompoziciju s pivotiranjem, stoga dajemo opis još jedne metode, DDMD_R4_QRCP. Zatim pokazujemo još jedan način kako, koristeći DMD metodu, možemo rekonstruirati promatranu dinamiku sustava, tj. kako rekonstruirati željene uzorke sustava (DMDSP). Naposljetku, opisane metode primijenjujemo na dinamičkom sustavu koji opisuje Navier-Stokesovu jednadžbu.

Abstract (english)

In this paper we study the DMD method as a tool for numerical analysis of nonlinear dynamical systems. The DMD method is data driven and equation free method, hence, with many applications in many fields of engineering. We analyze the mathematical background and show that DMD method extracts important structures from a high dimensional dynamical system – the so called coherent spatio-temporal structures. These structures are given in the eigenvectors of an operator A which describes given dynamical system. The operator A is mostly unknown, but, based on given measurements (observables) of the system, the DMD provides a linear approximation of the operator A, the matrix S r and later finds its eigendecomposition. DMD is based on the SVD of the matrix of the snapshots and Rayleigh-Ritz method for approximating the eigendecomposition. We show how to reconstruct dynamical system using DMD algorithm. We define Koopman operator and described its action on the observables. We briefly describe eigendecomposition of the Koopman operator and show its connection to the DMD. We show DMD application in video processing, that is, we show how to separate background and foreground of a video. We test DMD method on two datasets and compare given results with RPCA method, which is most commonly used nowadays for that purpose. In the last section we give variations of the DMD algorithm. We present how to determine parameter of dimension reduction, we analyze how to improve Rayleigh-Ritz pairs and describe two new algorithms: DDMD_R4 and DDMD_R4_C. We show how to approximate the operator A if we want for the approximation to reside in the span of selected snapshots, and we see we can apply the QR decomposition with column pivoting instead of the SVD, and in that way we obtain another method, DDMD_R4_QRCP. We also show another way of reconstructing dynamical system in application using the DMD method, that is, how to reconstruct wanted snapshots (DMDSP). As a final example, we apply the presented algorithms on a dynamical system given by the Navier-Stokes equation.

• Keywords: DMD metoda; numerička analiza nelinearnih dinamičkih sustava; Rayleigh-Ritzova metoda
• Keywords (english): DMD method; numerical analysis of nonlinear dynamical systems; Rayleigh-Ritz method
• Language: croatian
• URN:NBN: urn:nbn:hr:217:451109
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2020-12-02 11:15:28